Equation (REF ) determines the scaled amplitude \(\tilde{a}_{s}\)  of the solitary waves of the CDSW once
the scaled amplitude \(\tilde{a}_{r}\)  of the shed resonant radiation is known. In a previous analysis of the nematic DSW
a WKB solution for the shed resonant radiation was found which was found to be in excellent agreement with numerical
solutions [1]}. A WKB solution was used as the resonant wavetrain is short wave relative to the DSW. This
solution for the resonant wavetrain was given in terms of the original variables \(u\)  and \((x,z)\) . Returning the amplitude relation (REF )
to the original variables using \(a_{s} = \epsilon ^{2} \tilde{a}_{s} = (u_{i} - u_{+}) \tilde{a}_{s}\)  and
\(a_{r} = \epsilon ^{2}\tilde{a}_{r} = (u_{i} - u_{+}) \tilde{a}_{r}\)  and using the resonant wavetrain solution of [1]} gives that the amplitude of the CDSW solitary waves is
determined by
\(a_{s} \left[ 1 + 2 C_{4}a_{s} \right]= 2\left(u_{i} - u_{+}\right)\frac{1- \frac{3}{4} \frac{\left( c_{g}-\sqrt{\frac{2}{q}} \: u_{+} \right) a_{r}^{2}}{B_{2}(u_{i} - u_{+})^{3}}}{1 - \frac{1}{2} \frac{a_{r}^{2}}{(u_{i} - u_{+})^{2}}},\) 
